## Abstract

So far, we have discussed two extreme ends of fractal geometry. We have explored fractal monsters, such as the Cantor set, the Koch curve, and the Sierpinski gasket; and we have argued that there are many fractals in natural structures and patterns, such as coastlines, blood vessel systems, and cauliflowers. We have discussed features, such as self-similarity, scaling properties, and fractal dimensions shared by those natural structures and the monsters; but we have not yet seen that they are close relatives in the sense that maybe a cauliflower is just a ‘mutant’ of a Sierpinski gasket, and a fern is just a Koch curve ‘let loose’. Or phrased as a question, is there a framework in which a natural structure, such as a cauliflower, and an artificial structure, such as a Sierpinski gasket, are just examples of one unifying approach; and if so, what is it? Believe it or not, there is such a theory, and this chapter is devoted to it. It goes back to Mandelbrot’s book, *The Fractal Geometry of Nature*, and a beautiful paper by the Australian mathematician Hutchinson.^{2} Barnsley and Berger have extended these ideas and advocated the point of view that they are very promising for the encoding of images.^{3} In fact, this will be the focus of the appendix on image compression.

## Keywords

Target Image Fractal Geometry Hausdorff Distance Initial Image Encode Image## Preview

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## Reference

- 1.Michael F. Barnsley, Fractals Everywhere, Academic Press, 1988.Google Scholar
- 2.J. Hutchinson, Fractals and self-similarity, Indiana Journal of Mathematics 30 (1981) 713–747.MathSciNetMATHCrossRefGoogle Scholar
- J. Hutchinson, Some of the ideas can already be found in R. F. Williams, Compositions of contractions, Bol. Soc. Brasil. Mat. 2 (1971) 55–59.CrossRefGoogle Scholar
- 3.M. F. Bamsley, V. Ervin, D. Hardin, and J. Lancaster, Solution of an inverse problem for fractals and other sets, Proceedings of the National Academy of Sciences 83 (1986) 1975–1977.MathSciNetCrossRefGoogle Scholar
- M. Berger, Encoding images through transition probabilities, Math. Comp. Modelling 11 (1988) 575–577.CrossRefGoogle Scholar
- A survey article is: E. R. Vrscay, Iterated function systems: Theory, applications and the inverse problem, in: Bélair, J. and Dubuc, S., (eds.), Fractal Geometry and Analysis, Kluwer Academic Publishers, Dordrecht, Holland, 1991.Google Scholar
- A very promising approach seems to be presented in the recent paper A. E. Jacquin, Image coding based on a fractal theory of iterated contractive image transformations, to appear in: IEEE Transactions on Signal Processing. See also the chapter Fractal Image Compression by Y. Fisher, R. D. Boss, and E. W. Jacobs, to appear in Data Compression, J. Storer (ed.), Kluwer Academic Publishers, Norwell, MA.Google Scholar
- A similar metaphor has been used by Barnsley in his popularizations of iterated function systems (IFS), which is the mathematical notation for MRCMs.Google Scholar
- Almost any image can be used for this purpose. Images with certain symmetries provide some exceptions. We will study these in detail further below.Google Scholar
- 8.Thi:; explains the binary characterization of the Sierpinski gasket which we have used in chapter 3, page 175, for the discussion of self-similarity.Google Scholar
- 9.M. F. Barnsley, Fractal Modelling of Real World Images, in: The Science of Fractal Images, H.-O. Peitgen and D. Saupe (eds.), Springer-Verlag, New York, 1988, page 241.CrossRefGoogle Scholar
- 15.The computational problem evaluating the Hausdorff distance for digitized images is addressed in R. Shonkwiller, An image algorithm for computing the Hausdorff distance efficiently in linear time, Info. Proc. Lett. 30 (1989) 87–89.CrossRefGoogle Scholar
- 16.The algorithms discussed in the appendix on image compression try to circumvent this problem.Google Scholar
- 17.Similar concepts are in M. F. Barnsley, J. H. Elton, and D. P. Hardin, Recurrent iterated function systems, Constructive Approximation 5 (1989) 3–31.MathSciNetCrossRefGoogle Scholar
- M. Berger, Encoding images through transition probabilities, Math. Comp. Modelling 11 (1988) 575–577.CrossRefGoogle Scholar
- R. D. Mauldin and S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988) 811–829.MathSciNetMATHCrossRefGoogle Scholar
- G. Edgar, Measures, Topology and Fractal Geometry, Springer-Verlag, New York, 1990. The first ideas in this regard seem to be in T. Bedford, Dynamics and dimension for fractal recurrent sets, J. London Math. Soc. 33 (1986) 89–100.Google Scholar